1. Methods for Bayesian Inference This Specific Aim will be to continue development and application of Data Augmentation methodology. Recently, Ritter and Tanner (1992) have developed a grid-based approach to extend one such algorithm, the Gibbs Sampler, to the nonconjugate case. The overall foal will be to compare the performance of this Griddy-Gibbs algorithm to competing methods such as the Metropolis algorithm, as well as to hybrids of the Metropolis algorithm in the context of biostatistical problems. The first subaim is to use this methodology to allow one to sample from the partial likelihood and then to apply this methodology to a variety of problems related to the Cox model. The second subaim is the application of Data Augmentation methods to allow sampling from the likelihood or posterior density for the logistic regression model (with and without random effects). Problems for subaims one and two to be addressed include: parameter inference; patching missing (at random) covariates; and goodness-of-fit. 2. Semiparametric Regression Methodology for Censored Data This Specific Aim will be to continue development and application of methodology for the regression analysis of censored lifetime data, when one does not assume any particular parametric family of survival distributions. The multiple imputation approach presented in Wei and Tanner (1992) will be extended. The first subaim is to replace the least-squares estimate of the regression parameters by more robust estimates. The second subaim is to use nonparametric methodology to remove the linear assumption regarding the systematic component of the model. The third subaim will be an investigation f large-sample properties of these various estimators. The small sample Frequentist properties of these estimators will be examined via simulation. 3. Frequentist Calculations using Markov Chain Methods Barndorff-Nielsen (1983) has developed an approximation to the density of the maximum likelihood estimator. In this Specific Aim, Markov Chain methods (such as the Gibbs Sampler, the Griddy-Gibbs Sampler and the Metropolis algorithm) will be applied to this Barndorff-Nielsen formula to perform Frequentist inference. Thus, the focus is on the behavior of the estimator in repeated samples, rather than as in Specific Aim #1 where we focus on the distribution of the parameter given the data. These Markov chain methods will be used to obtain marginals of the density of the MLE, calculate tail areas (p-values), calculate the distribution of the likelihood ratio statistic and construct Frequentist confidence intervals. The methodology will be applied to problems in biostatistics. 4. Development of Software to Accompany Methodology The goal is to develop Data Augmentation software, as applied to such problems as parametric and semiparametric censored regression methodology, as well as to logistic and Cox regression.